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450 AN APPLICATION OF PELL'S EQUATION 2x2 = x2 '+ y2 - z2 [Dec. (mod p) . By definition, x is a quadratic residue of p. The above congruence implies 2x2 is also a quadratic residue of p. If p were of the form 8t ± 3, then 2 would be a quadratic nonresidue of p and since x2 is a quadratic residue of p, 2#2 would be a quadratic nonresidue of p, a contradiction. Thus p must be of the form St ± 1. Now, if we assume that there is a finite number of primes of the form St ± 1, and if we let m be the product of these primes, then we obtain a contradiction by imitating the above proof that there are infinitely many primes. 2 REFERENCE 1. W. Sierpinski. "Pythagorean Triangles." Soripta New York: Yeshiva University, 1964. Mathematica No. 9. Studies, AN APPLICATION OF PELL'S EQUATION DELANO P. WEGENER Central Michigan University, Mt. Pleasant, MI 48859 The following problem solution is a good classroom presentation or exercise following a discussion of Pellfs equation. Statement of the Problem Find all natural numbers a and b such that a (a + 1) = 2 h2 ° ' An alternate statement of the problem is to ask for all triangular numbers which are squares. Solution of the Problem a(a+ 1) = b 2^ a 2 + a = 2 b 2 ^ a 2 + a _ 2b2 = Q an odd integer t such that t2 ^ a - 2(2b)2 = -l ± /l + 8 & 2 ^ 3 = 1. This is Pell*s equation with fundamental solution [1, p. 197] t - 3 and 2b = 2 or, equivalently, t = 3 and b = 1. Note that t = 3 implies a = -1 ± 3 2 » but, according to the following theorem, we may discard a = -2. is odd. Also note that t TkZQtiQJfn 1: If D is a natural number that is not a perfect square, the Diophantine equation x2 - Dy2 = 1 has infinitely many solutions x9 y. All solutions with positive x and y are. obtained by the formula xn + yn/D = (x1 + y±/D)n where x1 9 y± is the fundamental solution of x all natural numbers. 2 - Dy , 2 = 1 and where n runs through 1981] CENTRAL FACTORIAL NUMBERS AND RELATED EXPANSIONS Z\(Xn) shows that all solutions ^n' A comparison of (xn + yjl) (3 + 2/2) and \\ XA of t 2 - 2(2ft)2 = 1 are obtained by V2 3/\22>J and hence all solutions of a ^ a 451 J/ \2bn+1) = £>2 are obtained from an = - ~ , bn = —j~» Note that tn is odd for all n so an is an integer. CENTRAL FACTORIAL NUMBERS AND RELATED EXPANSIONS Ch. A. CHARALAMBIDES University. of Athens, Athens, 1. Greece INTRODUCTION The central factorials have been introduced and studied by Stephensen; properties and applications of these factorials have been discussed among others and by Jordan [3], Riordan [5], and recently by Roman and Rota [4], For positive integer m9 x[m>b] = x(x + ~mb - b)(x + ±mb - 2b\ -.. (x - ~m£> + b\ defines the generalized central factorial of degree m and increment b. This definition can be extended to any integer m as follows: x['m'b] = x2/xlm+2,b] , m a positive integer. The usual central factorial (b = 1) will be denoted by x[m^. Note that these factorials are called "Stephensen polynomials" by some authors. Carlitz and Riordan [1] and Riordan [5, p. 213] studied the connection constants of the sequences x^m^ and xn9 that is, the"central factorial numbers t{m9 ri) and T(m3 ri): m m n = 0 ^=0 these numbers also appeared in the paper of Comtet [2], In this paper we discuss some properties of the connection constants of the sequences x^m'd] and x^-n'h^9 h ^ g9 of generalized central factorials, that is, the numbers K(jn9 n, s): m x[m'g] = Jl gmKnK(m9 rc = 0 2. n9 s)x[n> h \ s = h/g. EXPANSIONS OF CENTRAL FACTORIALS The central difference operator with increment a, denoted by Sa , is defined by &af(x) = f(x + a/2) - f(x - a/2) Note that E* - E^ =^" i A a s (2.1)